Some Computational Results
about Grid Diagrams of Knots
Yael Degany, Andrew Freimuth, and Edward Trefts
April 18, 2008
1
Introduction
From Heegaard Floer homology, an invariant for three-manifolds, one can
\
\
construct an invariant for knots- knot Floer homology HF
K. HF
K is
a finite dimensional bi-graded vector space over Z/2. One can introduce
^
as well HF
K, which is an invariant of grid diagrams dependent only on
the knot K and the arc index n of the diagram. It is a theorem from
N n−1
^
\
[MOS06] that there is an isomorphism HF
K = HF
K
V
where V
^
is a two dimensional Z/2 vector space. Thus from HF
K we can recover
\
^
HF
K which is a knot invariant independent of n. HF
K is related to the
Alexander polynomial ∆K (T ) by the formula:
(1 − t−1 )n−1 ∆K (T ) =
X
^
(−1)j ti dim HF
Ki,j
(1)
i,j
\
From [MOST06] it can be seen that HF
K can be arrived at by combinatorial means. We can do this by working with toroidal grid diagrams.
A toroidal grid diagram is a planar grid diagram where the top and bottom
edges are identified and the left and right edges are identified. A planar grid
diagram is a n × n grid where every row contains exactly one X and one O,
every column contains exactly one X and one O and no cell contains more
than one X or O.
We draw lines between X or O if they are in the same column or row.
If we have a crossing we let the vertical line pass over the horizontal.
We can associate a chain complex to a toroidal grid diagram.
The generators are given by the n-tuples of intersection points between
horizontal and vertical arcs (viewed as circles on the torus) on the diagram,
1
1
INTRODUCTION
2
X
O
X
O
X
O
X
O
O
X
Figure 1: A Grid Diagram for the Trefoil
with the added condition that every intersection point appears on a unique
horizontal and vertical circle . There is a straightforward one-to-one correspondence between generators of our complex and elements x ∈Sn . This
choice of labelling for the generators depends on how one cuts open the
torus.
We define for A,B two collections of points on the plane the number
I(A, B). I(A, B) counts the number of pairs (a1 , a2 ) ∈ A and (b1 , b2 ) ∈ B
with a1 < b1 and a2 < b2 .
Finally we define J(A, B) as the average of numbers I(A, B) and I(B, A).
We define the functions A(x) and M (x) as follows:
1
1
n−1
A(x) = J(x, X) − J(x, O) − J(X, X) + J(O, O) −
2
2
2
M (x) = J(x, x) − 2J(x, O) + J(O, O) + 1
(2)
(3)
^
^
In our chain complex CF
K we say the generator x ∈ CF
Kij if and only
if A(x)=i and M(x)=j.
^
There is a differential in our bigraded chain complex δ: CF
Kij →
2
^
CF Kij−1 . The differential satisfies δ = 0.
^
^
It is the case that HF
K is the homology of CF
K.
In this paper, we will use the combinatorial methods outlined by [MOST06]
2
LEMMAS ABOUT GRID DIAGRAMS
3
to compute the genus of all torus knots. We use the following statement
g(K) =
^
^
(highest Alexander grading in HF
K =
6 0) − (lowest Alexander grading in HF
K =
6 0) n − 1
−
2
2
where g(K) denotes the genus of a knot K. This is adapted from theorem
N n−1
^
\
1.2 of [OS04] and the isomorphism HF
K = HF
K
V
from [MOST06].
We are thankful to Professor Robert Lipshitz and Thomas Peters for
supervising our summer research. We were supported by funds from the
Columbia University Math Department. We worked closely with Jonathan
Hales, Dmytro Karabash, and Michael T. Lock. We also want to thank
Professor Peter Ozsvath.
2
Lemmas about Grid Diagrams
We say that a point on a grid diagram â = (k, ak ) has a weight w(â) = c − d
where c is the number of X = (a, b) such that a ≥ k, b ≥ ak , d is the number
of Xs such that k > a, ak > b
Lemma 1. On any grid diagram if we have w(â) = r, then b̂ = (k + 1, ak )
has weight r − 1, similarly ĉ = (k, ak + 1) has weight r − 1.
Proof. Say w(â) = r where â = (k, ak ). So (k + 1, ak ) is a shift over to
the right of the diagram. In the column separating the two points there
exists exactly one X = (a, b).There are two cases b ≥ ak or b < ak . In
first case we get w(k + 1, ak ) = (c − 1) − d = r − 1 and in the second
w(k + 1, ak ) = c − (d + 1) = r − 1. A similar argument can be made for ĉ,
where the only difference in the weight must be one X in the row separating
the points.
Figure 2 gives an example of a grid diagram with the weights filled
in. The distribution of the weights solely depends on the size of the grid
diagram; thus two n × n grid diagrams of two different knots will have the
same weight distribution.
Lemma 2. For any grid diagram I(x, X) − I(X, x) = n where x = Id.
Proof. Say that the âi are the points that make up the generator x. Then
I(x, X) − I(X, x) = w(â1 ) + w(â2 ) + ... + w(ân ) = n + (n − 2) + (n − 4)... +
P
n(n−1)
2
= n.
(n − 2(n − 1)) = n2 − 2 n−1
i=1 i = n − 2
2
Lemma 3. Let (ab) denote the transposition exchanging a and b. For a
generator x if I(x, X) − I(X, x) = n then I((ab)x, X) − I(X, (ab)x) = n.
3
COMPUTING THE GENUS
0
-1
4
-2
X
1
-4
-5
-3
-4
-1
-2
-3
-1
-2
-1
-3
O
-1
0
-2
X
O
2
1
0
3
2
1
0
4
3
2
1
0
5
4
3
2
1
X
O
X
O
O
X
0
Figure 2: The weights for a trefoil
Proof. Let y = (ab)x. Then the intersection points that x and y do not have
in common form a rectangle of length m on the grid. Let r be the weight of
the upper right point of the rectangle and s be the weight of the lower left
point where both belong to x. Then the upper left and lower right points r0
and s0 belong to y (See Figure 3). Then we have s0 = r+m and r0 = s−m. So
I(y, X)−I(X, y) = I(x, X)−I(X, x)−r −s+r0 +s0 = I(x, X)−I(X, x).
From these lemmas, it easily follows that for every grid diagram we
must have I(x, X) − I(X, x) = n and similarly I(x, O) − I(O, x) = n if x is a
generator in our chain complex. This is just an application of the well-known
fact that every permutation can be written as a product of transpositions.
This identity will be useful in simplifying the formulas for the Alexander
and Maslov gradings and as well in our computation of the genus of torus
knots.
3
Computing the genus
The genus of a p,q torus knot is
(p − 1)(q − 1)
2
It was also found in [OS04] that a knot’s maximal Alexander grading(the
highest Alexander grading which is nontrivial) is the genus. Using this result
COMPUTING THE GENUS
{
s
5
...
r
...
m
r’
...
3
...
s’
Figure 3: A detail for a grid diagram where the two generators x and y differ
by a transposition
and by finding general equations for the maximal and minimal Alexander
gradings of any p,q torus knot, we were able to confirm (1).
To find equations for the maximal and minimal Alexander gradings, it is
necessary to see that all p,q torus knots can be represented on a grid diagram
with grid number p+q [MB98], a top left to bottom right diagonal of X’s or
O’s, and parallel diagonals of the alternate (O’s or X’s) terminating at the
edges and of length p and q respectively. That this representation describes
any p,q torus knot can be seen by reversing the connection of X’s and O’s
so that there are no crossings on the grid diagram. The resulting diagram
will intersect one edge of the diagram p times and the other q times, as seen
in the center diagram in Figure 4.
The Alexander grading in a grid diagram with grid number n is defined
as
1
ni − 1
Ai (x) = J(x − (X + O), Xi − Oi ) −
2
2
1
1
n−1
= J(x, X) − J(x, O) − J(X, X) + J(O, O) −
2
2
2
where for A, B ⊆ R2 J(A, B) = I(A,B)+I(B,A)
where I(a, b) is the number of
2
pairs (a1 , a2 ) ∈ A and (b1 , b2 ) ∈ B such that a1 < b1 and a2 < b2 .
Only the first two terms of the equation are dependent on the specific
matching, and need to be considered for maximizing or minimizing the
3
COMPUTING THE GENUS
6
Figure 4: Representation of trefoil on toroidal grid diagram
Alexander gradings of a torus knot. These two terms can be simplified.
By using the identities I(x, X) − I(X, x) = n and I(x, O) − I(O, x) = n one
observes that minimizing or maximizing the Alexander gradings corresponds
to minimizing or maximizing the following quantities:
Ares (x) = I(x, X) − I(x, O)
or Ares (x) = I(X, x) − I(O, x)
It is not too difficult to see that the quantity Ares (x) corresponds to the sum
of winding numbers of each point on the matching with the knot oriented
from O’s to X’s in each column. In the general grid diagram for a torus
knot in Figure 5, Ares (x) = 0 for all matchings in areas a and d. This is
seen from the previous equations and because any point in the region a will
be above no X’s or O’s and in region d will be below no X’s or O’s. Arel (x)
will be negative for matchings in areas b and c because there are no X’s
and always O’s below any x in area b and in area c any x is always below
an O but never below any X. Therefore the maximum value of Arel (x) is 0,
and because of the structure of the grid diagram the unique matching where
this is possible is forced to be the matching with points in the upper left
corner of all O’s. Since the grid diagram is a torus, it can be shifted upwards
by a Dynnikov type 3 move, cyclic permutation, shown in Figure 6. Then
by similar reasons, the values of Arel (x) around the center diagonal will be
all positive, the values in the corners are all 0, forcing the unique matching
with the least relative Alexander grading to be the matching with points in
the upper left corner of all X’s.
As all torus knots will have a grid diagram of the same general form, and
the maximal and minimal Alexander grading can be computed in general
terms for a Tp,q torus knot in terms of p and q. Looking at each component
3
COMPUTING THE GENUS
Figure 5: A general form of a grid diagram for a p,q torus knot
Figure 6: A Dynnikov type 3 shift
7
REFERENCES
8
of A(x) in terms of p and q gives the following formulas
Amin (x) =
1h
(p + q) − (p + q)(p + 1)
2
min(p,q)
X
+2
i + (p + q − 1 − 2min(p, q))min(p, q)
i=1
+pq + p + q − 1
i
q
for p > q, Amax (x) =
X
1h
p(q − 1) + 2
i + q(p − q) − q(p + 1)
2
i=1
i
−p(q − 1) + pq − (p + q − 1)
p
for p < q, Amax (x) =
X
1h
p(q − 1) + 2
i + (q − p)(p + 1) − q(p + 1)
2
i=1
i
−p(q − 1) + pq − (p + q − 1)
These simplify to
Amin (x) =
Amax (x) =
−pq − p − q + 1
2
(p − 1)(q − 1)
pq − p − q + 1
=
= genus
2
2
References
[MB98]
H.R. Morton and E. Beltrami. Arc index and the kauffman
polynomial. Mathematical Proceedings of the Cambridge Philosophical Society, 123:41, 1998.
[MOS06]
Ciprian Manolescu, Peter Ozsvath, and Sucharit Sarkar. A combinatorial description of knot floer homology. 2006.
[MOST06] Ciprian Manolescu, Peter Ozsvath, Zoltan Szabo, and Dylan
Thurston. On combinatorial link floer homology. 2006.
[OS04]
Peter Ozsvath and Zoltan Szabo. Holomorphic disks and genus
bounds. Geom. Topol., 8:311, 2004.